I know of three, have only played one. That is of course if you are talking about steel stringed solid bodies with magnetic pick-ups.
I own a Risa "kidney bean". I absolutely adore it, but they are getting herder to find. Risa also makes a "Les Paul" steel string, and it is a LOT prettier than the kidney bean, and a lot more expensive. My kidney has a great tone and is perfect chimey surf style with it's lipstick pickups. The only thing I would wish for really would be a tapered neck (it seems pretty straight to me).
Here is a complete review of it .
Next up is a
Monkey Wrench- I was originally looking at these as they are BEAUTIFUL, but could not get a contact back- sent the email that said hey take my money and never got a reply, so I sent that money to the person who owned my Risa. That is neither here nor there- folks who have them like them a lot and like I said- pretty. Really pretty.
Last one is a
Jupiter Creek. I have never seen one of these or heard one played, but they look pretty darn good- fellow is a one man show though and availability is limited.
You have
these as well, but I have never heard of anyone who has owned one.
Aside from that, there is the idea of a really small electric guitar (tuned ADGCEA), 1/4 size would make a decent six string and I think Fender makes these.
There are also solid bodied nylon stringed electric ukes that use transducer pickups- Eleuke is the one to buy, steer clear of the Stagg (from what I have heard)- quick internet search will show you all kinds of these out there, but if you go this route, go
Mim's Ukes. Hope that helps- remember, a steel string ukulele is a whole different breed of cat- they feel and sound absolutely nothing like a regular uke.